This answer is not based on Nuzlocke Viability Rankings, as they slightly differ. It also doesn't take post-game content into consideration.
Rankings based on the analysis:
1. Alakazam
2. Jynx (Trade)
3. Mr. Mime (Trade)
Alakazam
Alakazam is undoubtedly the strongest Pokémon in FR/LG. With a sky-high Special Attack stat of 135 and a speed of 120, it sweeps a lot of gym leaders and E4 members. The only downside it has is its physical frailty, but that shouldn't hinder much when you can outspeed and OHKO most of the time. Due to Psychic-type being so broken even in FR/LG and the lack of Dark-type Pokémon before post-game, it can spam STAB Psychic to the fullest and doesn't need any coverage moves. Access to Calm Mind makes it a potent sweeper, taking down entire teams after a couple of boosts.
Since Abra evolves into Kadabra at level 16, it is possible to trade and evolve it into Alakazam right away. This makes the early and mid-game extremely easy.
Alakazam @ Twisted Spoon / Anything
Ability: Any
EVs: 252 SpA / 252 Spe / Any
Modest/Timid Nature
- Psybeam --> Psychic
- Calm Mind
- Psybeam (for higher PP) / Filler
- Filler
Jynx
Jynx, despite being available after getting access to fishing, is a trade Pokémon and hence gets boosted EXP; this allows it to quickly catch up with the rest of the team. It naturally learns Ice Punch which is special in this Gen; this can really helpful as the only other reliable way (outside of Game Corner TMs) of getting an Ice-type move is Ice Beam on Lapras. Access to Calm Mind and Psychic (both available for free) makes it a potent sweeper. It comes with a fixed Mild Nature, which is amazing for it as it increases SpA. Even though it has a bad typing and abysmal Defense, it has enough Speed to outspeed most threats. It can also learn Perish Song for when in need.
Jynx @ Never-Melt Ice / Anything
Ability: Oblivious (fixed)
EVs: 252 Spe / 252 SpA / Any
Mild Nature (fixed)
- Psychic
- Calm Mind
- Powder Snow --> Ice Punch / Ice Beam (if you're willing to grind for the TM, though there's only a 15 BP difference)
- Filler
Mr. Mime
Mr. Mime is available somewhat early, right after getting access to Diglett's Cave. Even though it will most likely be at a low level, its a trade Pokémon and gets boosted EXP. It has above average SpA and Speed stats, and is very bulky on the special side. It naturally learns a decent coverage move in Magical Leaf (which can't miss) and gets access to Calm Mind making it a potent sweeper. It can learn some TMs for coverage, though Psychic-type moves being so broken are easily spammable on their own.
Mr. Mime @ Twisted Spoon / Anything
Ability: Soundproof (fixed)
EVs: 252 SpA / 252 Spe / Any
Timid Nature (fixed)
- Psybeam --> Psychic
- Calm Mind
- Magical Leaf / Filler
- Filler
(will add more to this later)
Sources: Lots of experience playing FR/LG for the past 7 years, this and this
Hope this helps!